A short discussion of steady rectilinear plastic flow of Bingham material between rigid cylinders of general shape in relative motion parallel to their generators was given in a paper of the same main title(1). It was shown that, for sufficiently great relative velocity V of the boundaries, plastic flow in the whole region would be expected, as for a Newtonian liquid. But for slower relative motion, part of the material might remain elastically deformed (i.e. stressed below the yield point) with a limited region of plastic flow surrounding the inner boundary. In the paper referred to, the problem was illustrated by the case of eccentric circular cylindrical boundaries. The choice of circular boundaries introduced a great simplification when the region of plastic flow did not extend to the outer boundary, namely axial symmetry in the flow pattern in the plastic region. It is the main purpose of the present paper to illustrate how the methods already given for determining velocity distributions may be modified to suit more general problems, by consideration of confocal elliptic cylinders as boundaries.